How Many Days Equal A Year? Non-trivial on the Mean-Variance Model
George L. Ye, Dr.
Sobey School of Business
Saint Mary’s University
Halifax, Nova Scotia, Canada
Christine Panasian, Dr.
Sobey School of Business
Saint Mary’s University
Halifax, Nova Scotia, Canada
Abstract
In portfolio management, the means and variances of stock
returns are usually estimated on a daily basis and then converted
to longer periods of time. This paper examines the issue of how to
convert 1-day means for longer periods and investigates the
impacts of this conversion on capital allocation decisions and
portfolio performance evaluations.
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1. Introduction
The mean-variance model and the related efficient frontier provided the basis of
the development of modern portfolio theory.
It offered the first systematic analysis
of the risk-return trade-off faced by each investor and it still constitutes a fundamental
theory for portfolio formation.
Markowitzs’ ground-breaking theoretical model and
associated numerical solutions of portfolio selection under uncertainty supported the
foundation for many important developments in financial economics like the Capital
Asset Pricing Model (CAPM), derived by Sharpe (1964), Lintner (1965), and Mossin
(1966)) and the now classical distinction between systematic and idiosyncratic risks.
Although the importance of his work remains undeniable, mean-variance portfolio
analysis generated more questions than answers and, consequently, caused a large
body of research.
In this paper, we address the importance of an issue which is very practical, but
often ignored, on estimating the means and variances of stock returns. In practice,
mean and variance on a stock return is often estimated on a 1-day horizon, and then
they are converted to longer horizons as needed, by scaling. However, this conversion
is not always trivial, because the time scale used for this conversion may not be the
same as the calendar days and depends on the stock price movement.
First, in the literature, Fama (1965), French (1980), Roll (1984), French and Roll
(1986) and many other researchers find that volatility is caused by trading activities.
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This means that only trading days should be counted to determine the time scale. For
example, the 1-year variance of return on a stock traded on the New York Stock
Exchange (NYSE) should be equal to 1-day variance multiplied by 252, the number
of trading days in a year on NYSE, rather than by 365, as shown in Hull (2012).
Although some researchers, including Smithsonite and Minton (1996 a,b), J.P.
Morgan (1996), Diebold et al (1997), among others, show different perspectives, this
approach has been widely accepted by industry and regulators such as the Basel
committee.
However, little attention has been paid to the issue of converting a 1-day mean
to a longer time horizon. While the volatility is caused by trading activities, the mean
of a series of returns depends on a variety of factors among which time and the time
value of money consideration. This implies that non-trading days should also be
counted toward calculating the time scale. At the same time, since only trading days
contribute to volatility, or market price risk, the expected returns on no trading days
cannot be same as those on trading days. Thus, for instance, the simple rule of
multiplying by 252 may not be correct to convert a 1-day mean to 1-year mean.
In this paper, we present an approach to convert a 1-day mean to a longer horizon,
based on the Capital Asset Pricing Model (CAPM), and show how this conversion
may affect the decisions of capital allocation and on the portfolio performance
evaluation.
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2. Mathematical Framework
Let E(r x ) and σ x be the mean and standard deviation of the daily return on stock X,
i.e., 1-day mean and 1-day standard deviation, respectively. If the time horizon of our
investment is a period of N calendar days, in order to use the mean-variance model to
make optimal investment decisions, we need to adjust these parameters to fit our time
horizon.
We denote E*(r x ) and σ x * to be the mean and standard deviation of the return on
stock X for the time period of N calendar days, and n the number of trading days in
the period. Then, since only trading days contribute to volatility, we have:
σ *X = σ X
n
(1)
However, not only the number of trading days contributes to the determination of the
mean, but also that of the non-trading days. Thus, the mean for an N-day period is
consisting of two components: the total mean of n trading days and the total mean of
(N-n) non-trading days. The first component is the 1-day mean multiplied by the
number of trading days, i.e., nE(r x ). Regarding the second component, in the light of
the CAPM, the expected return consists of the risk free rate of return and a risk
premium, representing compensation for investment time and for the risk assumed,
respectively. During a non-trading day, an investment will require compensation for
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time value of money only. However, since risk or volatility is mainly caused by
trading activities, the risk on a non-trading day can be ignored, and the risk premium
required should be close to zero. Thus, the second component is equal to (N-n)R F ,
where R F is the daily risk free rate of interest. Therefore, in summary, we can
calculate the mean for an N-day period as follows:
E*(r x )= nE(r x )+ (N-n)R F
(2)
In some financial markets, such as foreign currency exchange market, trading is
carried out seven days a week, and therefore N, the number of calendar days, is the
same as n, the number of trading days. Since both the mean and standard deviation
are adjusted by the same time scale, the investment horizon should have no impact on
the portfolio decision.
However, in other financial markets, such as stock markets around the globe,
trading is interrupted during weekends and stock market holidays, and therefore,
number of trading days is less than the number of calendar days. For instance, there
are 252 trading days on the New York Stock Exchange in one year, that is n=252
while there are N=365 calendar days. Thus, formula to convert 1-day mean and
standard deviation to 1-year ones are:
E*(r x )= 252E(r x )+ 113R F
(3)
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σ *X = σ X
252
(4)
Since the mean and standard deviation are adjusted by different time scales, the
optimal portfolio selection may differ with different investment horizon. In next
section, we will show some examples on this effect.
3. Effects of Time Horizon Changes
3.1.
Effects on Capital Allocation
For the purpose of illustration, we present a numerical example as follows:
From the historical daily data from December 1, 2009, to December 1, 2012, we
obtained the estimates of the mean and standard deviation of the daily return on the
S&P 500 as below:
E(r S&P500 )= 0.000407
σ S&P500 = 0.011648
For simplicity assume the risk free rate is R f *=3% per annum, or R f =0.000082 per
day. 1
1
0.03/365=0.000082
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Following Bodie et al (2012), we assume an investor has a utility function as
follows:
U= E ( r) – (A/2) σ2
with a degree of risk aversion, A, equal to 4,
First, assuming that the investment horizon is one day, following Bodie et al.
(2012), the optimal capital allocation between the risk free asset and the equity for the
investor is:
y d = (E(r S&P500 )-R F )/(A σ S&P500 2) = (0.000407- 0.000082)/(4x0.0116482 )
=60%
This suggests investing 60% in the equity portfolio and the remaining 40% in T-Bills.
Now, assuming that the investment horizon is one year and using Eq. (3) and
(4), the annual mean and standard deviations are calculated as follows:
E*(r S&P500 )= 252x0.000407+113x0.000018=0.1046
σ* S&P500 = 0.011648x2521/2=0.1849
Similarly, by following Bodie et al.(2012) the optimal allocation between the risk free
asset and the equity is:
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y a = (E*(r S&P500 )-R* f )/(A σ* S&P500 2)
= (0.1046- 0.03)/(4x0.18492 )
=55%
This suggests investing 55% in the equity portfolio and the remaining 45% in T-Bills.
The above simple example shows that the investment horizon could affect capital
allocation decision considerably.
3.2.
Effect on Performance Measurement
On a daily basis, the Sharpe ratio of an asset is given by:
Sx = (E(rX )-RF)/σX
and on an annual basis, the Sharpe ratio is
S*x = (E*(rX )-R*F)/σ*X
Since
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E*(rX )-R*F= nE(rx )+ (N-n)RF -NRF =n(E(rX )-RF)
the annual Sharpe ratio becomes:
S*x = n(E(rX )-RF)/ σX n1/2
= n1/2 Sx
The above equation indicates that, though the value of the Sharpe ratio is dependent
on the investment horizon, the ranking among securities traded in the same market
will remain the same. For example, for two portfolios of NYSE traded stocks, if one
is better than the other based on the daily Sharpe ratio, the same ranking will be
maintained if the annual Sharpe ratio is used. However, when we compare the
performances of securities traded in different markets with different number of
trading days, the investment horizon may affect their ranking.
For the purpose of illustration, we can look at the following numerical
example:
Consider a stock X traded on NYSE and a foreign currency Y, traded on the
FX market. Assume that the means and standard deviations of the annual rate of
returns on X and Y are given as follows:
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Stock X:
E*(rX ) = 20% , σ*X =30%
Currency Y:
E*(rY ) = 12% , σ*Y =8%
The risk free interest rate is R*F = 4% per annum.
The number of calendar days is N=365, while the number of trading days for X is
n x =252 and that for Y is n y =365.
Then on an annual basis, the Sharpe ratios of X
and Y are as follows:
S*X = (E*(rX )-R*F)/σ*X =(0.2-0.04)/0.3=0.53
S*Y = (E*(rY )-R*F)/σ*Y = (0.12-0.04)/0.14=0.57
Since S* X <S* Y , we conclude that currency Y has better performance than stock X.
If we perform the same analysis on a daily basis, using equations (2) and (3),
the means and standard deviations of the daily rate of returns on X and Y are
calculated as follows:
Stock X:
E(rX ) = 0.000745, σX =0.018898
Currency Y:
E(rY ) = 0.000329, σY =0.007328
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The risk free interest rate is RF = 0.000110 per day.
Then the Sharpe ratios of X and Y are as follows:
SX = (0.000745-0.000110)/0.018898=0.0336
SY = (0.000329-0.000110)/0.007328=0.0299
Since S X > S Y , we conclude that stock X has better performance than currency Y.
The above example indicates that the difference in investment horizon may affect
the performance ranking of the assets.
One note worth mentioning in this context is that since beta is independent of
time horizon, investment horizon should not affect the performance ranking given by
the beta- based performance measures, such as Treynor ratio.
4. Conclusions and Suggestions
When using the mean-variance framework in portfolio management, it is common
to estimate the 1-day means and variances of stocks and other assets, and then convert
them to ones with longer horizons, such as annual means and variance. While the
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conversion of variances has been widely addressed and there is an approach popularly
accepted, the conversion of means has been rarely addressed.
In this paper, we present a formula to convert 1-day means to 1-year means or
means with other horizons, based on the CAPM model. Then, with this formula, we
investigate the impact of time horizon changes to the optimal capital allocation
decision and to the performance evaluation, with numerical examples.
We show that the optimal capital allocation decision can be different for a
particular investor if he or she would change his or her investment horizon. In
addition, when the Sharpe ratio is used to evaluate the performance of portfolios, the
choice of the time horizon may significantly affect the evaluation.
While we present a simple approach for the conversion of means with different
time horizons, the issue is much more complex. As we are working in the meanvariance framework, the expected return consists of 2 distinguishable components:
the risk-free rate and market risk premium. Thus, since market risk is mainly caused
by trading activities, risk premium on non-trading days can be safely ignored.
Consequently, the mean on a non-trading day should include only the risk-free
interest rate. However, many recent studies have shown that there are some other
factors which also contribute to mean levels. One of the most significant
developments in this direction of researches is the Liquidity-based Asset Pricing
Model (LAPM) developed by Amihud and Mendelson (1986), Acharya and Pedersen
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(2005), and Liu (2006), among others, which indicates that illiquidity is priced as a
security characteristic and/or as a risk factor and such, the illiquidity premium should
be added to the expected returns. In the light of the LAPM model, non-trading days
should have higher illiquidity premium than trading days, although they have lower
market risk premiums. How to incorporate these issues into the conversion formula
for means will be a direction of our future research.
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